is market order dynamics because as a market maker we only use limit orders which are matched and lled by market orders Given this model we can simulate order executions and run our market making strategy For simplicity we make the following assumptions 1 No latency 2 No price impact and 3 No competition with other market makers

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1 Introduction, Market makers are critical providers of liquidity in markets as they constantly place bid and. ask orders in the limit order book such that any market order will always be capable of being. filled The goal of the market maker is to strategically place these bids and asks to capture the. spread the difference between the bid and ask price while also earning a rebate for providing. liquidity On the other hand market making has become one of the prevailing strategies for. high frequency traders who profit by turning over positions in an extremely short period. These high frequency traders play integral roles in providing liquidity to markets accounting. for more than 50 of total volume in the US listed equities SEC 2014. Various pricing models for market making have been proposed in the academic literature Ho. and Stoll 1981 is one of the early studies that analyze the market making problem under a. stochastic control framework Avellaneda and Stoikov 2008 extends the model proposed by. Ho and Stoll 1981 derives the optimal bid and ask quotes using asymptotic expansion and. applies it to high frequency market making Furthermore Gue ant Lehalle and Fernandez. Tapia 2013 develops the model further by deriving the closed form solution of the optimal. bid and ask spread with boundary conditions on inventory size. In this paper we implement the high frequency market making pricing model proposed. by Avellaneda and Stoikov 2008 We choose this model for the ease of implementation. and analysis and unlike recent models such as Gue ant et al 2013 it does not restrict the. permissible inventory size Although this unconstrained inventory assumption lets the market. maker to keep trading regardless of their position it is a shortcoming since it increases the. likelihood of accumulating a one sided position and getting exposed to inventory risk the risk. that we bear due to inventory To complement the pricing model we develop an inventory. control model that dynamically adjusts the order size based on the current position This. integrated approach allows us to effectively control inventory risk while ensuring profitability. A trading simulator is devised to assess the P L and inventory of our optimal pricing strategy. compared to a baseline pricing model for five representative stocks. The rest of the paper is organized as follows Section 2 describes the pricing model and. the inventory model section 3 explains a trading simulator on which the strategy is tested. section 4 discusses the experiment and results while section 5 concludes. 2 Market Making Model, As a high frequency market maker we integrate the pricing framework proposed by Avellaneda. and Stoikov 2008 and a proprietary order size dynamic model The combination of an op. timal quote and a dynamic order size strategy allows us to effectively control inventory risk. and ensure profitability,2 1 Pricing, We use the optimal market making model developed by Avellaneda and Stoikov 2008 as our. bid and ask quote setting strategy The framework is based on a utility maximizing market. maker trading in a limit order book This section presents a brief summary of the model. We are interested in maximizing our expected exponential utility given our profit and loss at. terminal time T Assuming the risk free rate is zero and the mid price of a stock follows a. standard brownian motion dSt dWt with initial value S0 s and standard deviation. Avellaneda and Stoikov 2008 formulates the market maker problem as. u s x q t max Et e XT qT ST,where a b are the bid and ask spreads. is a risk aversion parameter,XT is the cash at time T. qT is the inventory at time T,ST is the stock price at time T. A few assumptions must be made before solving the stochastic optimal control problem. First it is important to model inventory as a stochastic process given that order fills are. random variables Therefore we can model,qt Nta Ntb. where Nta is the amount of stock sold,Ntb is the amount of stock bought. Based on this definition we can model cash as a stochastic differential equation in the form. dXt pa dNta pb dNtb,where pa pb are the bid and ask quotes. Avellaneda and Stoikov 2008 also provides a structure to the number of bid and ask exe. cutions by modeling them as a Poisson process According to their framework this Poisson. process should also depend on the market depth of our quote This is achieved through the. following expression,where is the market depth, This framework to model execution intensity will also prove useful in the design of our. trading simulator Avellaneda and Stoikov 2008 then continue to solve the stochastic. control problem using the following Hamilton Jacobi Bellman equation. 0 ut 2 uss max b b u s x s b q 1 t u s x q t,a a u s x s a q 1 t u s x q t. Then this nonlinear partial differential equation is solved using an asymptotic expansion for. a small inventory q This results in the pricing equations that are relevant to our algorithm. r s t s q 2 T t Indifference price,a b 2 T t ln 1,Spread around r s t. It is important to notice that since Avellaneda and Stoikov 2008 defines T as the terminal. time in which the trader optimizes its expected utility the spread equation can be seen as a. linear function of T t given by,where A is the slope of the spread equation. B is the closing spread when t T, If 0 the spread equation becomes a decreasing function of time The rationale behind this. optimal strategy is that tighter optimal spread enables the market marker to liquidate their. position before the market closes so as to avoid overnight risk To implement the framework. developed by Avellaneda and Stoikov 2008 we must compute our indifference price and set. an optimal spread around it given by these two equations We exploit the linearity of the. spread equation and our market data in order to adjust our spread to the best bid ask spread. dynamics The calibration strategy is explained in detail in the Experiments section of the. 2 2 Inventory, To complement the pricing model we develop a proprietary dynamic order size framework. Unlike the strategy followed by Gue ant et al 2013 who stops quoting if the inventory. reaches the maximum permissible level we are able to keep trading and earning rebates by. adjusting the order size based on our current position Our order size model is given by the. following equation,t if q t 0 max,t e if qt 0 t e if qt 0. where bid ask,t t are the bid and ask order size at time t. t is the maximum order size at time t,is a shape parameter. We select the shape parameter 0 005 to obtain dynamic order size model shown in. Figure 1 As we will later see this function is very effective at controlling inventory risk. The main reasoning behind its mechanics is that the framework controls inventory risk by. placing smaller order sizes in the direction of excess position accumulation. Buy 100 shares Sell 100 shares,Order Size,Sell at function Buy at function. 600 400 200 0 200 400 600,Figure 1 Dynamic order size function. 2 3 Algorithm, As a market maker we are interested in implementing an algorithm that places bid and ask. quotes in the limit order book at all times However we are aware that in very brief periods. we must hold one sided quotes for the sake of profitability This situation occurs when both. buy and sell orders are not filled at the same time interval. Therefore our strategy iterates as follows During the trading day we quote a bid and ask. spread if we have no orders in the limit order book If only one of these orders is filled we. wait for 5 seconds for the outstanding order to be executed If this does not happen then. we cancel the order and place new bid and ask quotes Finally whenever we have two orders. in the limit order book we update our quotes every second The summary of the trading. algorithm is shown in Algorithm 1,Algorithm 1 Market Making Algorithm. while current time end time do,if no orders in the book then. Quote bid and ask prices,else if 1 order in the book then. if current time execution time waiting time then, Cancel the outstanding order Quote new bid and ask prices. else if 2 orders in the book then,if current time quote time update time then. Cancel both order Quote new bid and ask prices,3 Trading Simulator. We build a trading simulator to assess our strategy The principal constituent of the simulator. is market order dynamics because as a market maker we only use limit orders which are. matched and filled by market orders Given this model we can simulate order executions. and run our market making strategy For simplicity we make the following assumptions 1. No latency 2 No price impact and 3 No competition with other market makers. 3 1 Market Order Dynamics, Let denote the depth of our quote in the order book We use a Time Inhomogeneous. Poisson Process to model the number of arrivals of market orders that are matched with the. limit order at the depth of,Nt P ois s ds, Analogous to the market making model the intensity function is assumed to be a product. of time and depth components in the following form. A piece wise linear bathtub shape is adopted for the time component t based on the empirical. result of intraday volume pattern as discussed in Cartea Jaimungal and Penalva 2015 The. depth component e is a decreasing function of depth since the deeper a quote is the. less likely it is for the order to be executed due to the lower priority Figure 3 illustrates the. shape of these two components respectively,a Time component t b Depth component e. Figure 2 The shape of intensity function,3 2 Order Execution. The market order dynamics enables us to simulate an order execution as follow At time. t we generate a Bernoulli random number X Ber t with the depth of a limit. order and time interval Then X 1 indicates the arrival of a market order and we. assume that the order is executed To make the experiment more realistic we also allow an. order to be partially filled by generating another random number from Gamma distribution. Y Gamma Then Y is multiplied by our order size to compute the executed order. size For instance Y 1 implies a full execution while Y 1 means a partial fill. 4 Experiments, We choose the week of June 12 2017 to simulate our strategy and trade from 9 30am 4pm. each day As the market maker may deal with a wide variety of stocks we choose the S P500. as a baseline and then four stocks to represent combinations of high and low volume as well. as high and low performance as shown in Table 1 With the technique described in section. 2 1 the parameters are calibrated using the average of the opening and closing spreads from. each stock in the previous week The data is retrieved from a simulator provided by Thesys. Technologies and the time interval is set to 1 second The maximum order size max t is. set to 100 The parameters of the simulator are set as 100 2 1 1 65. To assess the performance of the optimal strategy in section 2 we consider a baseline pricing. strategy in which the market maker always quotes at the best bid and ask prices that are. currently placed in the order book Every other aspect of the baseline strategy remains. identical to the optimal strategy Then a Markov Chain model is used to further examine. probabilities that help measure performance of these strategies. Table 1 Stocks to trade,Volume Performance Open Spread Close Spread. AAPL high high 0 05 0 01,AMZN low high 0 49 0 56,GE high low 0 04 0 01. IVV low high 0 03 0 01,M low low 0 09 0 01,4 1 Results. Our primary interest is the profitability and the inventory management of the strategies. Table 2 shows the average terminal P L and position inventory of both strategies The. optimal strategy achieves higher profits in AAPL and AMZN and comparable profits in. the rest of stocks compared with the baseline strategy Also both strategies end each. trading day with a small position on average indicating the success of inventory management. Furthermore the optimal strategy accomplishes the variance reduction of profits and position. per day as shown in Table 3 Not only does the optimal strategy reduce the terminal position. but it also manages the inventory during a trading session while consistently making profits. Table 2 Average terminal P L and position,Optimal Baseline. P L Position P L Position,AAPL 1 378 01 29 0 1 625 25 45 6. AMZN 58 331 04 8 4 13 522 2 34 6,GE 703 12 49 2 708 48 38 4. IVV 217 58 41 8 547 52 36 8,M 534 04 46 0 587 09 23 8. Table 3 The mean and standard deviation of profits and position per day. Optimal Strategy Baseline Strategy,Profits Position Profits Position. Mean Stdev Mean Stdev Mean Stdev Mean Stdev, AAPL 988 54 289 82 0 86 63 66 1093 60 357 66 7 53 112 2. AMZN 32 426 72 16 157 0 48 52 438 33 4 889 20 4 202 4 2 96 126 94. GE 245 0 192 92 2 41 60 92 248 96 191 43 11 97 109 19. IVV 23 14 129 9 0 49 67 9 152 0 196 6 1 38 109 59, M 144 26 146 78 0 83 46 14 192 59 167 24 3 86 105 93. Figure 3 demonstrates the market and optimal spreads for AAPL and GE on June 12 2017. Note that the spreads are rounded to the nearest cent As discussed in section 2 the spread. of the optimal pricing model is a linear function of time At the beginning the optimal. spread is wider than the market spread Later in the trading day around 2pm there is a. turning point when the optimal spread becomes narrower than the market spread meaning. that more orders are likely to be filled This aggressiveness helps unwind the accumulated. position before the market closes and boost our profits if spreads are large enough. a AAPL b GE,Figure 3 Market and optimal spreads, a Cumulative P L in the optimal strategy b Cumulative P L in the baseline strategy. c The density of inventory size in the optimal strategy d The density of inventory size in the baseline strategy. Figure 4 AAPL, Figure 4 displays P L and inventory size for APPL on each day in both strategies Though. we observe the poorest performance overall on AAPL we observe that the inventory size is. centered around 0 For the optimal strategy the position remains between 200 and the. density has sharper peak than that of the baseline strategy which suggests more control of. inventory risk In contrast to AAPL for GE P L consistently increases over time The same. trend of profits is observed in the baseline strategy but the variance of profits is lower in. the optimal strategy As is the case of AAPL the optimal strategy outperforms the baseline. strategy in terms of controlling the inventory size. We also compute the average number of buy and sell orders executed shares bought and sold. and quotes per day as shown in Table 4 and 5 For both strategies the number of buy and. sell orders as well as the number of shares bought and sold are well balanced indicating that. the position stays around 0 In terms of quoting the baseline strategy updates and posts. prices more frequently than the optimal strategy However the optimal strategy quotes more. efficiently because it has more buy sell orders executed per quote than the baseline strategy. a Cumulative P L in the optimal strategy b Cumulative P L in the baseline strategy. c The density of inventory size in the optimal strategy d The density of inventory size in the baseline strategy. Figure 5 GE, Table 4 The average number of orders shares and quotes per day in the optimal strategy. buy orders sell orders shares bought shares sold quotes. AAPL 6 085 6 105 446 921 447 066 64 645,AMZN 14 530 14 394 696 907 696 865 42 130. GE 4 411 4 456 324 148 324 394 82 138,IVV 7 613 7 625 548 505 548 714 63 349. M 2 927 3 003 221 074 221 304 91 292, Table 5 The average number of orders shares and quotes per day in the baseline strategy. buy orders sell orders shares bought shares sold quotes. AAPL 7 969 7 895 528 109 528 337 472 437,AMZN 7 833 7 850 500 537 500 710 528 028. GE 6 438 6 239 422 223 422 415 601 208,IVV 7 361 7 394 486 863 487 047 666 219. M 6 421 6 487 430 305 430 186 738 390,4 2 Markov Chain Analysis. We apply a Markov Chain model to compute the probabilities that measure the performance. of strategies Let us assume that the state space S consists of 0 1 2 Quoting W aiting Spread. as depicted in Figure 6 Quoting means bid and ask prices are quoted W aiting means one. of the orders is filled and the market maker is waiting for the outstanding order to be filled. and Spread means both buy and sell orders are filled and the spread is captured Note that. the transition probability from Spread to Quoting is 1 and there is no arrow from Spread. to W aiting because after making the spread the market maker always quotes new bid and. ask prices Table 6 summarizes the interpretation of each transition probability. p 0 0 Quoting Waiting p 1 1,Figure 6 The Markov Chain model. Table 6 The interpretation of the transition probabilities. p 0 0 Update bid ask prices,p 0 1 One of the orders is filled. p 1 0 Cancel the outstanding order after waiting,p 1 1 Wait for the outstanding order to be filled. p 0 2 Both buy sell orders are filled, p 1 2 The outstanding order is filled after waiting. A quantity of interest for market makers is the probability of capturing the spread since it. is the main source of profit in the business There are two scenarios in which the market. maker can capture the spread in each quote The first case is that both buy and sell orders. are filled within the next time interval The other case is that one of the order is filled and. after 5 waiting periods the outstanding order is filled Therefore the probability of making. the spread is,p p 0 2 p 0 1 p 1 1 n p 1 2, Another quantity of interest would be the probability of one sided fill which is the case where. only buy or sell is filled and the remaining order is cancelled after the waiting time The. higher likelihood of one side fill increases the inventory risk since the fill is imbalanced The. probability of one side fill after 5 waiting periods is defined as. q p 0 1 p 0 1 p 1 1 5 p 1 0,Table 7 The probability of making the spread. AAPL AMZN GE IVV M,Optimal strategy 2 6 19 3 1 8 4 9 0 9. Baseline strategy 5 1 4 7 3 7 4 7 3 8,Table 8 The probability of one side fill. AAPL AMZN GE IVV M,Optimal strategy 0 8 1 9 0 4 0 7 0 3. Baseline strategy 0 9 1 0 0 5 0 7 0 5, The estimated probabilities are summarized in Table 7 and 8 The baseline strategy has. higher probability of making the spread than the optimal strategy because the baseline. strategy always quotes at the best bid and ask prices which tend to be more aggressive than. the optimal prices Yet it is important to note that the higher probability of making the. spread may not always be indicative of the higher profit as it also depends on the size of. spreads The optimal strategy on the other hand achieves lower probability of one side fill. This finding demonstrates that the optimal strategy can quote more efficiently and prevent. the position from accumulating,5 Conclusions, In this project we implement the high frequency market making pricing strategy proposed. by Avellaneda and Stoikov 2008 The inventory strategy that mitigates the inventory risk is. proposed to complement the pricing model Furthermore we develop the trading simulator to. experiment with our strategy on real high frequency data With the inventory control model. the optimal pricing model achieves more controlled inventory size while ensuring profitability. A possible extension of our strategy can be to use a model to predict mid price and market. order arrivals so that the market makers can profit regardless of the market movement and. the size of spreads,References, Avellaneda M Stoikov S 2008 High frequency trading in a limit order book. Quantitative Finance 8 3 217 224 Retrieved from https doi org 10 1080. 14697680701381228, Cartea A Jaimungal S Penalva J 2015 Algorithmic and high frequency trading. Cambridge United Kingdom Cambridge University Press. Gue ant O Lehalle C A Fernandez Tapia J 2013 Dealing with the inventory risk a. solution to the market making problem Mathematics and Financial Economics 7 4. 477 507 Retrieved from https doi org 10 1007 s11579 012 0087 0. Ho T Stoll H R 1981 Optimal dealer pricing under transactions and return uncer. tainty Journal of Financial Economics 9 47 73, U S Securities and Exchange Commission 2014 Equity Market Structure Literature Re. view Part II High Frequency Trading,Appendix Results for All Stocks. Optimal Strategy on June 12 2017,a P L b Position,c Quoted Prices d Optimal bid ask spread. Figure 7 AAPL,a P L b Position,c Quoted Prices d Optimal bid ask spread. Figure 8 AMZN,a P L b Position,c Quoted Prices d Optimal bid ask spread. Figure 9 GE,a P L b Position,c Quoted Prices d Optimal bid ask spread.